where \(\tilde{V}_Z\equiv V_Z+\frac{U}{2}(\langle n_\uparrow \rangle -\langle n_\downarrow \rangle )\) , \(\tilde{\epsilon }\equiv \epsilon _0+\frac{U}{2}(\langle n_\uparrow \rangle +\langle n_\downarrow \rangle )\)  and \(\tilde{\Gamma }_S\equiv \Gamma _S+U \langle d_\uparrow d_\downarrow \rangle \) , with
\(\langle n_{\sigma } \rangle =\langle d^\dagger _\sigma d_\sigma \rangle \)  and \( \langle d_\uparrow d_\downarrow \rangle \) , being the QD spin-polarised occupations and the anomalous averages, respectively. \(\tilde{V}_Z\) , \(\tilde{\epsilon }\)  and \(\tilde{\Gamma }_S\)  can be physically interpreted, respectively, as an exchange field, a level shift and an effective coupling which are renormalized due to correlations (for a full derivation see the SI and also Ref. [1]}). Note that, for large \(U\) , the finite spin polarization \(\langle n_\uparrow \rangle \ne \langle n_\downarrow \rangle \)  that is induced at the singlet-doublet crossing explains the deviations from the pure Zeeman regime governed by \(V_Z\) . The same happens for large \(\Gamma _S\) . It is also worth nothing that the criterion for parity crossings of YSR states in Eq. (REF ) is essentially the same as the criterion for topological superconductivity in a proximitized nanowire by just substituting the renormalised QD parameters by the equivalent ones in a nanowire, ie. the external Zeeman field, the chemical potential and the local superconducting pairing: \(\tilde{V}_Z \rightarrow V^{NW}_Z\) , \(\tilde{\epsilon }\rightarrow \mu ^{NW}\)  and \(\Gamma _S\rightarrow \Delta ^{NW}\)  [2]}, [3]}. This is yet another example that emphasises the difficulty of making a clear distinction between parity crossings in QDs and MZMs in nanowires.
The LP modulation of the gap adds further complexity to the problem (since the ratio \(\Delta (B)/U\)  evolves with magnetic field), which results in a singlet-doublet transition in the first lobe at \(B = 105 \, {mT}\)  (red curves in Fig.  REFH) with little resemblance to the original Zeeman-split lines (green lines in Fig.  REFG). Together with the tunneling broadening \(\Gamma _N\) , this can result in a robust zero bias anomaly in the LDOS across the first LP lobe (white regions in Fig.  REFH). Our theoretical analysis shows that the observed ZBPs are actually the result of an intriguing magnetic field evolution (the full phase diagrams at three different \(B\) -fields are shown in Fig.  REFI).
